Geodesic
On the rate of convergence in Kolmogorov's uniform limit theorem.~I
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 225-245

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Theorem. {\it For any probability distribution function $F$ on $R$ and for any natural number $n$ there exists an infinitely divisible distribution function $B$ such that $$ \sup_x|F^{n*}(x)-B(x)|\le C_n^{-2/3} $$ } Here $F^{n*}$ is the $n$-fold convolution of $F$ with itself and $C$ is an absolute constant. The paper contains the first part of the proof. 
@article{TVP_1981_26_2_a0,
     author = {T. V. Arak},
     title = {On the rate of convergence in {Kolmogorov's} uniform limit {theorem.~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {225--245},
     publisher = {mathdoc},
     volume = {26},
     number = {2},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a0/}
}
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T. V. Arak. On the rate of convergence in Kolmogorov's uniform limit theorem.~I. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 225-245. http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a0/